On topological n-ary semigroups. 1. Introduction

Transcription

1 Quasigroups and relaed sysems, 3 (1996), 7388 On opological n-ary semigroups Wiesªaw A. Dudek and Vladimir V. Mukhin Absrac In his noe some we describe opologies on n-ary semigroups induced by families of deviaions. 1. Inroducion Topological ngroups were invesigaed by many auhors. For example, ƒupona proved in [5] ha each opological ngroup can be embedded ino a opological group. šiºovi described opological medial ngroups (cf. [0]), opological ngroups wih he Baire propery (cf. [1]) and proved a opological analog of Hosszú heorem (cf. [19]). Crombez and Six described a fundamenal sysem of open neighborhoods of a xed elemen (cf. [4]). Endres proved ha every opological ngroup is homeomorphic o some canonical opological group (cf. [9]). Topologies induced by norms are considered by Boujuf and Mukhin (cf. [] ). Balci Dervis ( cf. [1] ) described free opological ngroups. In [1] is described a mehod of embedding opological abelian nsemigroups in opological ngroup. On he oher hand, we known ha opological nsemigroups have many properies which are no rue for binary semigroups. In his paper we invesigae opologies on nsemigroups and n groups deermined by families of lef invarian deviaions. We describe 1991 Mahemaics Subjec Classicaion: 0N15, A30 Keywords: n-ary semigroup, n-ary group, opological semigroup, deviaion

2 74 W. A. Dudek and V. V. Mukhin he condiions under which such opology is compaible wih he n ary operaion. We nd also he necessary and sucien condiions for he opologically embedding a semiabelian opological nsemigroup in a opological ngroup.. Preliminaries Tradiionally in he heory of n-ary groups we use he following abbreviaed noaion: he sequence x i,..., x j is denoed by x j i (for j < i his symbol is empy). If x i+1 =... = x i+k = x, hen insead of xi+1 i+k we wrie (k) x. Obviously (0) x is he empy symbol. In his noaion he formula f(x 1,..., x i, x i+1,..., x i+k, x i+k+1,..., x n ), where x i+1 =... = x i+k = x, will be wrien as f(x i 1, (k) x, x n i+k+1). If m = k(n 1) + 1, hen he m-ary operaion g given by 1 ) = f(f(..., f(f(x n }{{} 1), x n 1 n+1 ),...), x k(n 1)+1 k imes g(x k(n 1)+1 (k 1)(n 1)+ ) will be denoed by f (k). In cerain siuaions, when he ariy of g does no play a crucial role, or when i will dier depending on addiional assumpions, we wrie f (.), o mean f (k) for some k = 1,,... An nary operaion f dened on G is called associaive if f(f(x n 1), x n 1 n+1 ) = f(x i 1 1, f(x n+i 1 i ), x n 1 n+i ) holds for all x 1, x,..., x n 1 G and i = 1,,..., n. The se G ogeher wih one associaive operaion f is called an nary semigroup (briey: nsemigroup). An nsemigroup (G, f) in which for for all a 1, a,..., a n, b G here exis an uniquely deermined x i G such ha f(a i 1 1, x i, a n i+1) = b is called an ngroup. From his deniion i follows ha a group (a semigroup) is a - group (a semigroup) in he above sense. Moreover, i is worhwhile o noe ha, under he assumpion of he associaiviy of f, i suces only o posulae he exisence of a soluion of he las equaion a

3 On opological n-ary semigroups 75 he places i = 1 and i = n or a one place i oher han 1 and n (cf. [13], p ). This means ha an ngroup may be considered as an algebra (G, f, f 1, f n ) wih one associaive nary operaion f and wo nary operaions f 1, f n such ha f(f 1 (a n, b), a n ) = f(a n a, f n (a n, b)) = b (1) for all a n, b G. Following E.L.Pos ([13], p.8) he soluion of he equaion f(x, a,..., a, f(a,..., a)) = a is denoed by a [ ]. An nsemigroup (G, f) wih an unary operaion [ ] : G G saisfying some naural ideniies is an ngroup (cf. [16]). The map x f(a j 1 1, x, a n j+1) is called an jh nary ranslaion deermined by a 1,..., a n. In an ngroup each nary ranslaion is a bijecion. In an ngroup (G, f) for any sequence a n 1 here exiss only one a G such ha f(x, a n 1, a) = f(a n 1, a, x) = f(a, a n 1, x) = f(x, a, a n 1 ) = x for all x G (cf. [17]). An elemen a is called inverse for a n 1. In he binary case, i.e. in he case n =, when he sequence a n 1 is empy by he inverse we mean he neural elemen of a group (G, f). A sequence a n is called a lef (righ) neural sequence if f(a n, x) = x (respecively f(x, a n ) = x) holds for all x G. A lef and righ neural sequence is called a neural sequence. In an ngroup for every sequence a n 1 may be exended o a neural sequence, bu here are nsemigroups wihou lef (righ) neural sequences. Le (G, f) be an nsemigroup and le a n 1 be xed. Then (G, ), where x y = f(x, a n 1, y) () is a semigroup, which is called a binary rerac of (G, f) and is denoed by re a n 1(G, f). A binary rerac of an ngroup is a group. Moreover, all binary reracs of a given ngroup are isomorphic (cf. [7]), bu n groups wih he same rerac are no isomorphic, in general.

4 76 W. A. Dudek and V. V. Mukhin By so-called Hosszú heorem (cf. [11] or [7]), every ngroup (G, f) has he form f(x n 1) = x 1 β(x ) β (x 3 )... β n 1 (x n ) b, (3) where a n is a xed righ neural sequence of (G, f), (G, ) = re a n 1(G, f), b = f( a (n) n ) and β(x) = f(a n, x, a n 1 ). The idenical resul holds for nsemigroups wih a righ neural sequence. 3. Topology An nsemigroup (G, f) dened on a opological space (G, T ) is called a opological nsemigroup if he operaion f is coninuous in all variables ogeher. A opological ngroup is dened as a opological nsemigroup wih wo addiional coninuous operaions f 1 and f n saisfying (1) (cf. [5]). A opological ngroup may be dened also a opological nsemigroup wih addiional coninuous operaion [ ]. These deniions are equivalen (cf. [15]). I is clear ha reracs of a opological nsemigroup (ngroup) are opological semigroups (groups). Obviously all ranslaions of a opological nsemigroup (ngroup) are coninuous maps. On he oher hand, every nary operaion which may by wrien in he form (3), where and β are coninuous, is coninuous in all variables ogeher. Thus he following lemma is rue. Lemma 3.1. Assume ha an nsemigroup (G, f) wih a opology T has a righ neural sequence a n. Then (G, f, T ) is a opological n semigroup if and only if re a n 1(G, f) is a opological semigroup and β(x) = f(a n, x, a n 1 ) is coninuous. Corollary 3.. An ngroup (G, f) dened on a opological space (G, T ) is a opological ngroup if and only if here exiss a righ neural sequence a n such ha x y = f(x, a n 1, y), β(x) = f(a n, x, a n 1 ) and [ ] : x x [ ] are coninuous.

5 On opological n-ary semigroups 77 Proposiion 3.3. An ngroup (G, f) dened on a opological space (G, T ) is a opological ngroup if and only if here exiss a righ neural sequence a n such ha re a n 1(G, f) is a opological semigroup, β(x) = f(a n, x, a n 1 ) and s : x s(x), where f(s(x), a n 1, x) = a n, are coninuous. Proof. Le a n be a xed righ neural sequence on an ngroup (G, f). If (G, ) = re a n 1(G, f) is a opological semigroup and β(x) = f(a n, x, a n 1 ) is coninuous, hen (G, f) is a opological nsemigroup by Lemma 3.1. Moreover, a n is he neural elemen of (G, ) and s(x) is he soluion of f(s(x), a n 1, x) = a n, i.e. s(x) x = a n in (G, ). Thus s(x) is he inverse of x in (G, ). Hence (G, ) is a opological group, because s(x) is coninuous, by he assumpion. Since f(z, c n ) = f(f(z, a n ), c n ) = z f(a n, c n ) for all c j G, hen he soluion z of f(z, c n ) = b in (G, f) is he soluion of z f(a n, c n ) = b in (G, ), hen z coninuously depends on b and f(a n, c n ). Thus z is a coninuous funcion of variables b, c,..., c n. This, for b = c =... = c n 1 = x, c n = f(x,..., x), implies ha z = x [ ] is a coninuous funcion of x. Thus (G, f) is a opological ngroup. The converse is obvious. Corollary 3.4. Le T be a locally compac opology on an ngroup (G, f) wih a righ neural sequence a n. If for every b G ranslaions x f(x, a n 1, b), x f(b, a n 1, x) and x f(a n, x, a n 1 ) are coninuous, hen (G, f, T ) is a opological ngroup. Proof. In he group (G, ) = re a n 1 (G, f) ranslaions x x b and x b x are coninuous for every b G. Thus, by he heorem of Ellis (cf. Theorem 3 in [8]), (G, ) is a opological group. In his group s(x) dened in he previous Proposiion is a coninuous operaion. Hence (G, f) is a opological ngroup.

6 78 W. A. Dudek and V. V. Mukhin 4. Deviaions By a deviaion dened on a nonempy se X we mean every map ϕ : X X [0, + ) such ha ϕ(x, x) = 0, ϕ(x, y) = ϕ(y, x), and ϕ(x, y) ϕ(x, z) + ϕ(z, y) for all x, y, z X. A deviaion ϕ dened on a semigroup (group) (G, ) is lef invarian if ϕ(cx, cy) = ϕ(x, y) for all c, x, y G. A deviaion ϕ dened on an nsemigroup (G, f) is a lef invarian if for all x, y, c n 1 1 G. ϕ(f(c n 1 1, x), f(c n 1 1, y)) = ϕ(x, y) Theorem 4.1 ([]). A binary semigroup (group) (G, ) wih a opology T is a opological semigroup (group) if and only if here exiss a family Φ of coninuous lef invarian deviaions on G which induces T and ϕ z Φ for every z G and ϕ Φ, where ϕ z is dened by ϕ z (x, y) = ϕ(xz, yz). In he case of an nsemigroup (G, f) every deviaion ϕ on (G, f) induces a new deviaion (ϕ, k, c n ) dened by (ϕ, k, c n )(x, y) = ϕ(f(c k, x, c n k+1), f(c k, y, c n k+1)), where c n G and k = 1,..., n are xed. Theorem 4.. Le a n be a righ neural sequence of an nsemigroup (G, f). If a opology T on G is induced by he family Φ of deviaions such ha for all x, y, z G and ϕ Φ (a) ϕ(f(z, a n 1, x), f(z, a n 1, y)) = ϕ(x, y), (b) (ϕ, 1, a n 1, z), (ϕ,, a n, a n 1 ) Φ, hen (G, f) is a opological nsemigroup. Proof. Le Φ be as in he assumpion. By (a) every ϕ Φ is a lef invarian deviaion on a semigroup (G, ) = re a n 1(G, f). From (b) we obain ϕ z (x, y) = ϕ(x z, y z) = ϕ(f(x, a n 1, z), f(y, a n 1, z)) =

7 On opological n-ary semigroups 79 = (ϕ, 1, a n 1, z)(x, y) for every z G, which gives ϕ z Φ. By Theorem 4.1 (G, ) is a opological semigroup. Le ε > 0. If x, x 0 G are such ha (ϕ,, a n, a n 1 )(x, x 0 ) < ε, where ϕ Φ, hen ϕ(β(x), β(x 0 )) = ϕ(f(a n, x, a n 1 ), f(a n, x 0, a n 1 )) = = (ϕ,, a n, a n 1 )(x, x 0 ) < ε, which proves ha β is coninuous. Lemma 3.1 nish he proof. Theorem 4.3. An ngroup (G, f) wih a opology T is a opological ngroup if and only if here exiss he family Φ of deviaions such ha a opology T is induced by Φ and for some righ neural sequence a n of G and for all x, y, z G, ϕ Φ he condiions (a), (b) from he previous heorem are saised. Proof. Le (G, f, T ) be a opological ngroup. Then he rerac (G, ) = re a n 1(G, f) is a binary opological group for every choice of a,..., a n 1 G. Thus, by Theorem 4.1, here exiss he family Φ of coninuous lef invarian deviaions of (G, ) which induces he opology T. Hence, for all x, y, z G and ϕ Φ, we have ϕ(f(z, a n 1, x), f(z, a n 1, y)) = ϕ(z x, z y) = ϕ(x, y), which proves (a). Moreover, since for all a,..., a n 1 G here exisa a n G such ha a n is a righ neural sequence, hen from he above follows ϕ(f(c n 1 1, x), f(c n 1 1, y)) = = ϕ(f(c n 1 1, f(a n, a n 1, x)), f(c n 1 1, f(a n, a n 1, y))) = = ϕ(f(f(c1 n 1, a n ), a n 1, x)), f(f(c n 1 1, a n ), a n 1, y))) = ϕ(x, y) for all c 1,..., c n 1 G. Thus every ϕ Φ is a lef invarian deviaion of an ngroup (G, f). Hence also (ϕ, k, c n ) is a lef invarian deviaion for every k = 1,,..., n and all c 1,..., c n 1 G. Obviously (ϕ, k, c n ) is

8 80 W. A. Dudek and V. V. Mukhin also lef invarian on (G, ) and (ϕ, k, c n ) Φ. Therefore (ϕ, 1, a n ), (ϕ,, a n, a n 1 ) Φ, which proves (b). Conversely, if a opology T is induced by he family Φ of deviaions saisfying (a) and (b), hen, by Theorem 4.1, (G, ) = re a n 1(G, f) is a binary opological group. Similarly as in he proof of Theorem 4. from (ϕ,, a n, a n 1 ) Φ follows ha he ranslaion β(x) = f(a n, x, a n 1 ) is coninuous. Proposiion 3.3 complees he proof. 5. Embedding of opological nsemigroups The necessary and sucien condiions for he embedding of opological semigroup in opological group are described by N. J. Rohman (cf. [14]) and F. Chrisoph (cf. [3]). In his secion we give some generalizaions of hese resuls. As i is well known (cf. for example [13] or [6]) an nsemigroup (G, f) is called semiabelian or (1, n)commuaive if f(x, a n 1, y) = f(y, a n 1, x) holds for all x, y, a,..., a n 1 G, and cancellaive if f(a i 1 1, x, a n i+1) = f(a i 1 1, y, a n i+1) = x = y for all i = 1,,..., n and x, y, a 1,..., a n G. Every ngroup is obviously cancellaive. Now we use he consrucion of he quoien ngroup presened during he Gomel's algebraic conference (1995) by A. M. Gal'mak and V. V. Mukhin. Le (G, f) be a cancellaive semiabelian nsemigroup. Then he relaion x, y z, f () ( (n 1) z ) = f () ( (n 1) dened on G G is an equivalence relaion. Indeed, he reexiviy and symmery are obvious. We prove he ransiiviy. Le x, y z, and z, u, v. Then f () ( (n 1) z ) = f () ( (n 1) x ) and f () ( (n 1) x ) u ) = f () ( (n 1) v z ).

9 On opological n-ary semigroups 81 Hence f (3) ( (n 1) x, (n 1) v ) = f (3) ( (n 1) z, (n 1) v ) = f (3) ( (n 1) v z ) = = f (3) ( (n 1) u ) = f (3) ( (n 1) u ),, (n 1) which by he cancellaiviy gives f () ( (n 1) x v ) = f () ( (n 1) u ). Since (G, f) is semiabelian, hen and in he consequence f () ( (n 1) x v ) = f () ( (n 1) v x ), f () ( (n 1) v x ) = f () ( (n 1) u ), which proves he ransiiviy. In he se G = G G/ of all equivalence classes x i, y i we dene he new nary operaion f ( x 1, y 1, x, y,..., x n, y n ) = f(x n 1), f(y n 1 ). If x i, y i s i, i for all i = 1,,..., n, hen also and f(f () ( (n 1) y 1 s 1 ),..., f () ( (n 1) y n f () ( (n 1) y i s i ) = f () ( (n 1) i s n )) = f(f () ( (n 1) 1 x i ) x 1 ),..., f () ( (n 1) n x n )). Bu every semiabelian nsemigroup is also medial ( see [10] ), i.e. i saises f(f(x 1n 11), f(x n 1),..., f(x nn n1)) = f(f(x n1 11), f(x n 1),..., f(x nn 1n)). Then he las ideniy may be wrien in he form which proves ha ( (n 1) f () f(y1 n (n) ), f(s n 1) ) ( (n 1) = f () f( n (n) 1), f(x n 1) ), f(x n 1), f(y n 1 ) f(s n 1), f( n 1).

10 8 W. A. Dudek and V. V. Mukhin Hence he operaion f is well dened. I is clear ha his operaion is also associaive and (1, n)commuaive. and Now le ( (n 1)(n ) x = f ( ) a, d, (n 1)(n 1) ) c ( (n 1)(n 1) y = f ( ) b, d, (n 1)n ) c, where a, b, c, d are xed elemens from G. Then, using (1, n)commuaiviy, we obain ( (n 1) f ( ) f(y, d ),..., f(y, (n 1) d ) a ) = }{{} (n 1) imes ( (n 1)(n 1) = f ( ) b, d, (n 1)n c, (n 1) (n 1)(n 1) d,..., b, d, (n 1)n c, (n 1) d a ) = {{}}{{}{{} (n 1) imes and ( (n 1) f ( ) b, f(x, (n 1) c ),..., f(x, (n 1) ) c ) = }{{} n imes ( (n 1) = f ( ) b a, (n 1) n d, (n 1) n ) c = W1 ( (n 1) f ( ) b, a, (n 1)(n 1) d, (n 1)(n ) c, (n 1) (n 1)(n 1) c,..., a, d, (n 1)(n ) c, (n 1) ) c {{}}{{}{{} n imes Since W 1 = W, hen ( (n 1) f ( ) f(y, d ),..., f(y, (n 1) d ) }{{} (n 1) imes which proves ha i.e. ( (n 1) = f ( ) b a, (n 1) n d, (n 1) n c ) = W. a ) ( (n 1) = f ( ) b, f(x, (n 1) c ),..., f(x, (n 1) ) c ) }{{} n imes f(x, (n 1) c ), f(y, (n 1) d ) = a, b,

11 On opological n-ary semigroups 83 f ( x, y, c, d,..., c, d ) = a, b. }{{} n 1imes Hence for all a, b, c, d G he las equaion has he soluion x, y G. In he similar way we prove ha for all a, b, c, d G here exiss x, y G such ha f ( c, d,..., c, d, x, y ) = a, b. }{{} (n 1) imes This proves (cf. [18]) ha (G, f ) is a semiabelian ngroup. The map p(x) = x, x is a homomorphic embedding of an n semigroup (G, f) in an ngroup (G, f ). Indeed, p(f(x n 1)) = f(x n 1), f(x n 1) = = f ( x 1, x 1,..., x n, x n ) = f (p(x 1 ),..., p(x n )) and p(x) = p(y) implies x, x = y, y, i.e. f () ( (n 1) x y ) = f () ( (n 1) x ) = f () ( (n 1) x, (n 1) y, x), which by he cancellaiviy gives x = y. Thus he following lemma is rue. Lemma 5.1. Every semiabelian cancellaive n-semigroup may be embedded ino a semiabelian n-group. Lemma 5.. If ϕ is a lef invarian deviaion of a cancellaive semiabelian nsemigroup (G, f), hen ϕ G ( x, y, z, ) = ϕ(f () ( (n 1) x ), f () ( (n 1) z ) ) is a lef invarian deviaion on G such ha ϕ G (p(x), p(y)) = ϕ(x, y). Proof. From he deniion of ϕ G follows ϕ G ( x, x, x, x ) = 0 and ϕ G ( x, y, z, ) = ϕ G ( z,, x, y ). Moreover, if x, y u, v, where x, y, u, v G G, hen f () ( (n 1) v x ) = f () ( (n 1) u )

12 84 W. A. Dudek and V. V. Mukhin and ϕ G ( x, y, z, ) = ϕ(f () ( (n 1) x ), f () ( (n 1) = ϕ(f (3) ( (n 1) v, (n 1) x ), f () ( (n 1) v, (n 1) = ϕ(f (3) ( (n 1), (n 1) v x ), f (3) ( (n 1) v, (n 1) = ϕ(f (3) ( (n 1), (n 1) u ), f (3) ( (n 1) v, (n 1) = ϕ(f (3) ( (n 1) u ), f (3) ( (n 1) v = ϕ(f () ( (n 1) u ), f () ( (n 1) v ϕ G ( u, v, z, ) which proves ha ϕ G is well dened. Now, for all x, y, z, G G we have ϕ G ( x, y, z, ) = ϕ(f () ( (n 1) x ), f () ( (n 1) ϕ(f (3) ( (n 1) v, (n 1) = ϕ(f (3) ( (n 1) v, (n 1) x ), f (3) ( (n 1) x ), f (3) ( (n 1) v, (n 1) z ) ) u ) ) + ϕ(f (3) ( (n 1) = ϕ(f (3) ( (n 1), (n 1) v x ), f (3) ( (n 1), (n 1) u ), f (3) ( (n 1) v, (n 1) u ) ) + ϕ(f (3) ( (n 1) u ), f (3) ( (n 1) v = ϕ(f () ( (n 1) v x ), f () ( (n 1) u ) ) + ϕ(f () ( (n 1) u ), f () ( (n 1) v = ϕ G ( x, y, u, v ) + ϕ G ( u, v, z, ). Hence ϕ G is a deviaion on G. To prove ha ϕ G is lef invarian observe ha for all i = 1,..., n 1, and a i, b i, a n 1, x, y, u, v G we have ϕ G ( f( a1, b 1,..., a n 1, b n 1, x, y ), f( a 1, b 1,..., a n 1, b n 1, u, v ) )

13 On opological n-ary semigroups 85 = ϕ G ( f(a n 1 1, x), f(b n 1 1, y), f(a n 1 1, u), f(b n 1 1, v) ) = = ϕ ( f () ( f(b n 1 1, v),..., f(b n 1 1, v), f(a n 1 1, x),..., f(a n 1 1, x) ), }{{}}{{} (n 1) imes n imes f () ( f(b n 1 1, y),..., f(b n 1 1, y), f(a1 n 1, u),..., f(a1 n 1, u) ) ). }{{}}{{} (n 1) imes n imes By he associaiviy and (1, n)commuaiviy of f, he las formula may be wrien in he form ϕ ( f (.) (..., (n 1) v x ), f (.) (..., (n 1) u ) ), which, ogeher wih he fac ha ϕ is lef invarian, implies ϕ ( f () ( (n 1) v x ), f () ( (n 1) u ) ) = ϕ G ( x, y, u, v ). This proves ha ϕ G is a lef invarian deviaion on G. Moreover ϕ G (p(x), p(y)) = ϕ G ( x, x, y, y ) = ϕ ( f () ( (n 1) x ), f () ( (n 1) x y ) ) = ϕ ( f () ( (n 1) x, x), f () ( (n 1) x, (n 1) y, y) ) = which complees our proof. = ϕ ( f () ( (n 1) x, x), f () ( (n 1) x, y) ) = ϕ(x, y), Theorem 5.3. A cancellaive semiabelian nsemigroup (G, f) wih a opology T may be opologically embedded in a opological ngroup if and only if a opology T is induced by a some family of lef invarian deviaions dened on G. Proof. If a cancellaive semiabelian nsemigroup (G, f) wih a opology T is opologically embedded in a opological ngroup (H, f) wih a opology T H, hen T H is induced by some family Φ of deviaions such ha ϕ(f(z, a n 1, x), f(z, a n 1, y) ) = ϕ(x, y),

14 86 W. A. Dudek and V. V. Mukhin where x, y, z H and a,..., a n is a righ neural sequence of an n group H (Theorem 4.3). Since in an ngroup H for all a,..., a n 1 H here exiss a n H such ha a,..., a n is a righ neural sequence, hen in he above formula all x, y, z, a,..., a n 1 are arbirary. This proves ha all ϕ Φ are lef invarian deviaions. Conversely, if a opology T on a cancellaive semiabelian nsemigroup (G, f) is induced by a some family Φ of lef invarian deviaions, hen every ϕ G dened in Lemma 5. is a lef invarian deviaion on G. By Theorem 4.3 he family {ϕ G } ϕ Φ induces on G he opology T G such ha G is a opological ngroup and p(x) = x, x is a opological embedding of (G, f, T ) in (G, f, T G ). References [1] Balci Dervis: Zur Theorie der opologischen n-gruppen, Minerva Publikaion, Munich [] H. Buzhuf, V. V. Mukhin: Topologies on semigroups and groups dened by families of deviaions and norms, (Russian) Izv. Vyssh. Uchebn. Zaved. Ma. 5 (1997), 74 77, (ransl. in Russian Mah. (Iz. VUZ 41 (1997), No. 5, 71 74)). [3] F. Chrisoph: Embedding opological semigroups in opological groups, Semigroup Forum 1 (1970), [4] G. Crombez, G. Six: On opological n-groups, Abhand. Mah. Semin. Univ. Hamburg 41 (1974), [5] G. ƒupona: On opological n-groups, Bull. Soc. Mah. Phys. R.S.Macédoine (1971), [6] W. A. Dudek: Remarks on n-groups, Demonsraio Mah. 13 (1980), [7] W. A. Dudek, J. Michalski: On a generalizaion of Hosszú heorem, Demonsraio Mah. 15 (198),

15 On opological n-ary semigroups 87 [8] R. Ellis: Locally compac ransformaion group, Duke Mah. J. 4 (1957), [9] N. Endres: On opological n-groups and heir corresponding groups, Discussiones Mah., Algebra and Sochasic Mehods 15 (1995), [10] K. Gªazek, B. Gleichgewich: Abelian n-groups, Colloquia Mah. Soc. J. Bolyai 9 "Universal Algebra", Eszergom (Hungary) 1997, (Norh-Holland, Amserdam 198). [11] M. Hosszú: On he explici form of n-group operaions, Publ. Mah. Debrecen 10 (1963), [1] V. V. Mukhin, H. Boujuf: On embedding n-ary abelian opological semigroups in n-ary opological groups, (in Russian) Voprosy Algebry 9 (1996), [13] E. L. Pos: Polyadic groups, Trans. Amer. Mah. Soc. 48 (1940), [14] N. J. Rohman: Embedding of opological semigroups, Mah. Ann. 139 (1960), [15] S. A. Rusakov: On wo deniions of opological n-ary groups, (Russian) Voprosy Algebry 5 (1990), [16] S. A. Rusakov: Algebraic n-ary sysems, (Russian), Izd. Navuka, Minsk 199. [17] F. M. Sioson: On free abelian m-groups, I, Proc. Japan. Acad. 43 (1967), [18] V. I. Tyuin: On he axiomaics of n-ary groups, Dokl. Acad. Nauk BSSR 9 (1985), [19] M. R. šiºovi : Topological analog of Hosszú-Gluskin's heorem, (Serbian), Ma. Vesnik 13 (8) (1976), [0] M. R. šiºovi : Topological medial n-quasigroups, Proc. Algebraic Conference, Novi Sad 1981, p

16 88 W. A. Dudek and V. V. Mukhin [1] M. R. šiºovi, Lj.D.Ko inac: Some remarks on n-groups, Zb. rad.fil. fak. u Ni²u, ser. Ma. (1988), Received May 10, 1997 W.A.Dudek V.V.Mukhin Insiue of Mahemaics Deparmen of Mahemaics Technical Universiy Technological Universiy Wybrze»e Wyspia«skiego 7 Sverdlova sr. 13 a Wrocªaw Minsk Poland Belarus dudek@im.pwr.wroc.pl or Higher College of Engineering ul. Jaworzy«ska Legnica Poland